It is, of course, when
our words describe the world that they are true. So for example, ‘Telly is
bald’ was a true description of Aristotelis Savalas when he was a baby. (As he
himself said, “We’re all born bald, baby.”) Now, Telly did not go from being a
bald baby to not being bald by growing just a few hairs, because ‘bald’ has not
got so precise a definition. So if, as seems possible, Telly did not suddenly
grow a lot of hair, then he will only gradually have stopped being bald. There
could, possibly, have been times when ‘Telly is bald’ was true, later times
when ‘Telly is bald’ was not true, and times in between when something else was
the case – or could there? If ‘Telly is bald’ was neither true nor untrue at
those intermediate times, then ‘Telly is bald’ was not true and was true, which
is ruled out by the meaning of ‘not’.
So, such intermediate
times seem to be logically impossible. And yet, we can hardly know a priori that Telly suddenly grew a lot
of hair. And while we can introduce new terms that are less vague than ‘bald’ –
e.g. 100 hairs or less and you are bald101, otherwise you are not – that
would hardly solve our problem with ‘bald’. So let us assume, for the sake of
argument, that Telly stopped being bald gradually: What was going on at the
intermediate times? Well, some of those around Telly may have been thinking of
him as bald, while others thought of him as not bald. And the vagueness of
‘bald’ gives us no reason to think that any of them were wrong. But, Telly was
certainly not very bald at such times, and nor was he clearly not bald, so why
not think of him as having been about as bald as not? Were ‘is bald’ about as
true as not of Telly, ‘Telly is bald’ would not so much not be true as be only about as true as not, and it would not so
much not be untrue as be about as
true as not. So, that would solve our problem.
We reason best with
descriptions that are either true or else not true, but the words of natural
languages are a little vague,1 so the two classical truth-values,
‘true’ and ‘false’,2 meet at a place – in logical space – where
descriptions are described as well by ‘not true’ as by ‘true’. For another
example, imagine a rough table-top being gently sanded flatter and flatter.
Eventually it becomes flat enough to count as flat, in the usual contexts. But
sanding away just a few scratches would hardly have flattened it, so the
borderline between flat and not flat is more like a pencil line than a
mathematical line. Our table-top will, briefly, be only vaguely flat, or about
as flat as not. ‘Flat’ is not, in that sense, well defined: It is a vague
predicate, not a definite predicate. But, there is a sense in which it is
defined perfectly well: There are such things as tables, which are flat by
design; and there are, similarly, bald men. Precisely redefining ‘flat’ and
‘bald’ – and ‘man’ and ‘table’ – in order to avoid the problem of vagueness
would lose us some of our ability to refer to reality. Indeed, we would lose
rather a lot of that basic function of language, because most of our words are
to some extent vague.
This also solves such
puzzles as the Sorites: We might suppose, for example, that the truth-value of
‘the table-top is flat’ could not change with the sanding away of a single
scratch. If so, then gently sanding a rough table-top for even a very long time
could not make true ‘the table-top is flat’. But, while ‘not flat’ is
contradicted by ‘flat’, it is not necessarily contradicted by ‘about as flat as
not’. So as the table-top begins to be about as flat as not, we would not be
wrong to call it ‘not flat’. Our calls could change from ‘not flat’ to ‘about
as flat as not’ in the blink of an eye, with no sanding at all. (Our original
supposition is less plausible when there is a borderline truth-value.)
There seems to be a ubiquitous
vagueness in natural language, but it is not really a problem. It is
surprising, but only because it is so unproblematic that it usually goes
unnoticed. Our words are defined as precisely as our purposes have required
them to be, and the slight vagueness means that we can always make them more
precise. When ‘Telly is bald’ becomes problematic, for example, ‘Telly is
getting hairy’ will be more straightforwardly true. ‘Getting hairy’ is hardly
less vague than ‘bald’, but its borderlines are in different places. We can
usually move the borderlines out of the way, even though we cannot remove all
the vagueness. And since we do not have to do much with descriptions that are
about as true as not – other than identify them as needing to be replaced with
truer descriptions – hence we need only adjoin ‘about as true as not’ to the
classical truth-values. Indeed, we should
only do that: The precision of formal logic is inapposite when we have left
behind the sharp division between something being the case and it not being the
case. A more formal definition could only be an inaccurate – if deceptively
precise – mathematical model of the most natural definition.
Following Aristotle,
the classical definitions are as follows. To say of what is the case that it is
the case, or of what is not the case that it is not the case, that is to speak
truly. And to say of what is the case that it is not the case, or of what is
not the case that it is the case, that is to speak falsely. So an adequate
adjunct could be: To say of what is about as much the case as not that it is
the case, or that it is not the case, that is to say something that is about as
true as not. A description that is much truer than not will be true enough to
count as true, by definition of ‘much’, while one that is not much truer than
not will be about as true as not by definition of ‘about’. And if we need to
make sharper distinctions, then we need to avoid borderline cases and use
classical logic.3
Now, descriptions are
normally of other things, but self-description is allowed – e.g. ‘this is in
English’ is a true self-description – so consider this example: This description is true. Let us call
that self-description ‘T’ (for Truth-teller). T says only that T is true, so it
is certainly possible for T to be true; but another possibility is that T is
false, because if T was false then it would follow from the meaning of T (that
T is true) only that T was not true. And since there is no more to T than that
– since T does nothing but describe itself (as true) – hence there is no reason
why T should be true rather than not true, or false rather than not false. So
it would make sense were T about as true as not.
Furthermore, some self-descriptions are paradoxical if they are not about as true as not (the post below concerns the Liar Paradox).
Notes
1. Bertrand Russell,
‘Vagueness’, The Australasian Journal of Psychology
and Philosophy 1 (1923), 84–92,
reprinted in Rosanna Keefe and Peter Smith (eds.), Vagueness: A Reader (Cambridge, MA: MIT Press, 1997), 61–68. For
the state of the art, see Richard Dietz and Sebastiano Moruzzi (eds.), Cuts and Clouds: Vagueness, Its Nature and
its Logic (New York and Oxford: Oxford University Press, 2010).
2. ‘“X is Y” is false’
just means that X is not Y, so in classical logic, where either X is Y or else
X is not Y, ‘false’ and ‘not true’ are interchangeable.
3. A good introduction
to the mathematics of classical logic is Stewart Shapiro, Classical Logic.